the function of measurement in modern physical science, thomas kuhn

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The Function of Measurement in Modern Physical Science By Thomas S. Kuhn* A T the University of Chicago, the facade of the Social Science Research „ Building bears Lord Kelvin’s famous dictum: “If you cannot measure, your knowledge is meager and unsatisfactory.”1 Would that statement be there if it had been written, not by a physicist, but by a sociologist, political scientist, or economist? Or again, would terms like “meter reading” and “yardstick” recur so frequently in contemporary discussions of epistemology and scientific method were it not for the prestige of modern physical science and the fact that measurement so obviously bulks large in its research? Suspecting that the answer to both these questions is no, I find my assigned role in this con- ference particularly challenging. Because physical science is so often seen as the paradigm of sound knowledge and because quantitative techniques seem to provide an essential clue to its success, the question how measurement hits actually functioned for the past three centuries in physical science arouses more than its natural and intrinsic interest. Let me therefore make my gen- eral position clear at the start. Both as an ex-physicist and as an historian of physical science I feel sure that, for at least a century and a half, quantitative methods have indeed been central to the development of the fields 1 study. On the other hand. I feel equally convinccd that our most prevalent notions both about the function of measurement and about the source of its special efficacy arc derived largely from myth. Partly because of this conviction and partly for more autobiographical rea- sons,2 I shall employ in this paper an approach rather different from that of most other contributors to this conference. Until almost its close my essay will include no narrative of the increasing deployment of quantitative tech- niques in physical science since the close of the Middle Ages. Instead, the two • University of California, Berkeley was added to the present program at a late 1 For the facade see, Eleven Twenty-Six: date, are abstracted from my essay, “The Role A Decaile of Social Science Research, ed. of Measurement in tfie Development of Nat- Louis W irth (Chicago, 1940), p. 169. The ural Science,” a multilithed revision of a talk sentiment there inscribed recurs in Kelvin’s first given to the Social Sciences Colloquium writings, but I have found no formulation of the University of California, Berkeley. That eloser to the Chicago quotation than the fol- version will be published in a volume of papers lowing: “When you cannot express it in num- on “Quantification in the Social Sciences” that bers, your knowledge is of a meagre and un- grows out of the Berkeley colloquium. In de- satisfactory kind.” See Sir William Tljomson, riving the present paper from it, I have pre- “Electrical Units of Measurement," Popular pared a new introduction and last section, and Lectures and Addresses, 3 vols. (London, 1889- have somewhat condensed the material that 91), I, 73. intervenes. 2 The central sections of this paper, which 161

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Page 1: The Function of Measurement in Modern Physical Science, Thomas Kuhn

The Function of Measurement in Modern

Physical ScienceB y Thomas S. K uhn*

AT the University of Chicago, the facade of the Social Science Research „ Building bears Lord Kelvin’s famous dictum: “ If you cannot measure,

your knowledge is meager and unsatisfactory.”1 Would that statem ent be there if it had been written, not by a physicist, but by a sociologist, political scientist, or economist? Or again, would terms like “ meter reading” and “yardstick” recur so frequently in contemporary discussions of epistemology and scientific method were it not for the prestige of modern physical science and the fact that measurement so obviously bulks large in its research? Suspecting that the answer to both these questions is no, I find my assigned role in this con­ference particularly challenging. Because physical science is so often seen as the paradigm of sound knowledge and because quantitative techniques seem to provide an essential clue to its success, the question how measurement hits actually functioned for the past three centuries in physical science arouses more than its natural and intrinsic interest. Let me therefore make my gen­eral position clear at the start. Both as an ex-physicist and as an historian of physical science I feel sure that, for a t least a century and a half, quantitative methods have indeed been central to the development of the fields 1 study. On the other hand. I feel equally convinccd tha t our most prevalent notions both about the function of measurement and about the source of its special efficacy arc derived largely from myth.

Partly because of this conviction and partly for more autobiographical rea­sons,2 I shall employ in this paper an approach rather different from that of most other contributors to this conference. Until almost its close my essay will include no narrative of the increasing deployment of quantitative tech­niques in physical science since the close of the Middle Ages. Instead, the two

• U niversity of California, Berkeley was added to the present program a t a late1 F or the facade see, E leven T w e n ty -S ix : date, a re abstracted from m y essay, “The Role

A Decaile o f Social Science Research, ed. of M easurem ent in tfie Development of N at-Louis W irth (Chicago, 1940), p. 169. T he ural Science,” a m ultilithed revision o f a talksentim ent there inscribed recurs in Kelvin’s first given to the Social Sciences Colloquiumw ritings, but I have found no form ulation of the U niversity of California, Berkeley. T hateloser to the Chicago quotation than the fol- version will be published in a volume of paperslow ing: “W hen you cannot express it in num- on “Q uantification in the Social Sciences” thatbers, your knowledge is of a m eagre and un- grow s out of the Berkeley colloquium. In de-satisfactory kind.” See S ir W illiam Tljomson, riv ing the present paper from it, I have pre-“E lectrical U nits of M easurem ent," Popular pared a new introduction and last section, andLectures and Addresses, 3 vols. (London, 1889- have somewhat condensed the m aterial tha t9 1 ), I, 73. intervenes.

2 T he central sections of this paper, which

161

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162 THOMAS S. KUIIN

central questions of this paper—how has measurement actually functioned in physical science, and what has been the source of its special efficacy— will be approached directly. For this purpose, and for it alone, history will truly be “philosophy teaching by example.”

Before perm itting history to function even as a source of examples, we must, however, grasp the full significance of allowing it any function a t all. To that end my paper opens with a critical discussion of what I take to be the most prevalent image of scientific measurement, an image that gains much of its plausibility and force from the manner in which computation and meas­urement enter into a profoundly unhistorical source, the science text. T hat discussion, confined to Section I below, will suggest that there is a textbook image or myth of science and that it may be systematically misleading. M eas­urement’s actual function—either in the search for new theories or in the confirmation of those already at hand— must be sought in the journal litera­ture, which displays not finished and accepted theories, but theories in the process of development. After that point in the discussion, history will nec­essarily become our guide, and Sections I I and I I I will attem pt to present a more valid image of measurement’s most usual functions drawn from that source. Section IV employs the resulting description to ask why measurement should have proved so extraordinarily effective in physical research. Only after that, in the concluding section, shall I attem pt a synoptic view of the route by which measurement has come increasingly to dominate physical sci­ence during the past three hundred years.

[One more caveat proves necessary before beginning. A few participants in this conference seem occasionally to mean by measurement any unambigu­ous scientific experiment or observation. Thus, Professor Boring supposes that Descartes was measuring when he demonstrated the inverted retinal image a t the back of the eye-ball; presumably he would say the same about Franklin’s demonstration of the opposite polarity of the two coatings on a Leyden jar. Now I have no doubt that experiments like these are among the most significant and fundamental tha t the physical sciences have known, but I see no virtue in describing their results as measurements. In any case, that terminology would obscure what are perhaps the most im portant points to be made in this paper. I shall therefore suppose that a measurement (or a fully quantified theory) always produces actual numbers. Experiments like Descartes’ or Franklin’s, above, will be classified as qualitative or as non- numerical, without, I hope, a t all implying tha t they are therefore less im­portant. Only with th a t distinction between qualitative and quantitative avail­able can I hope to show that large amounts of qualitative work have usually been prerequisite to fruitful quantification in the physical sciences. And only if that point can be made shall we be in a position even to ask about the effects of introducing quantitative methods into sciences that had previously proceeded without m ajor assistance from them.]

I. TEX TB O O K M EA SU REM EN TTo a very much greater extent than we ordinarily realize, our image of physi­

cal science and of measurement is conditioned by science texts. In part that

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MEASUREMENT IN MODERN PHYSICAL SCIENCE 163

influence is direct: textbooks are the sole source of most people’s firsthand acquaintance with the physical sciences. Their indirect iniluence is, however, undoubtedly larger and more pervasive. Textbooks or their equivalent are the unique repository of the finished achievements of modern physical scien­tists. It is with the analysis and propagation of these achievements that most writings on the philosophy of science and most interpretations of science for the nonscientist arc concerned. As many autobiographies attest, even the research scientist does not always free himself from the textbook image gained during his first exposures to science.8

I shall shortly indicate why the textbook mode of presentation must in­evitably be misleading, but let us first examine tha t presentation itself. Since most participants in this conference have already been exposed to at least one textbook of physical science, I restrict attention to the schematic tripartite summary in the following figure. I t displays, in the upper left, a series of

Theory(x ) 0i (x) Manipulation(x) 02(x) (Logic and M ath)

theoretical and “ lawlike statements, (x) 0 i(x), which together constitute the theory of the science being described.4 The center of the diagram represents the logical and mathematical equipment employed in manipulating the theory.“Lawlike” statements from the upper

3 T h is phenomenon is examined in more detail in my m onograph, T he S truc ture of Scientific Revolutions, to appear when com­pleted as Vol. IT. No. 2, in the International Encyclopedia o f Unified Science. M any other aspects of the textbook image of science, its sourccs and its strengths, a rc also examined in tha t place.

left are to be imagined fed into the4 Obviously not all the statem ents required

to constitute most theories a rc of th is particu­la r logical form , but the com plexities have no relevance to the points made here. R . B. B raithw aite, Scientific Explanation (C am ­bridge, England, 1953) includes a useful, though very general, description of the logi­cal s tructu re of scientific theories.

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hopper a t the top of the machine together with certain “initial conditions” specifying the situation to which the theory is being applied. The crank is then turned; logical and mathematical operations are internally performed; and numerical predictions for the application at hand emerge in the chute at the front of the machine. These predictions are entered in the left-hand col­umn of the table that appears in the lower right of the figure. The right-hand column contains the numerical results of actual measurements, placed there so tha t they may be compared with the predictions derived from the theory. M ost texts of physics, chemistry, astronomy, etc. contain many data of this sort, though they are not always presented in tabular form. Some of you will, for example, be more familiar with equivalent graphical presentations.

The table a t the lower right is of particular concern, for it is there that the results of measurement appear explicitly. W hat may we take to be the sig­nificance of such a table and of the numbers it contains? I suppose that there are two usual answers: the first, immediate and almost universal; the other, perhaps more important, but very rarely explicit.

M ost obviously the results in the table seem to function as a test of theory. If corresponding numbers in the two columns agree, the theory is acceptable; if they do not, the theory must be modified or rejected. This is the function of measurement as confirmation, here seen emerging, as it does for most read­ers, from the textbook formulation of a finished scientific theory. For the time being I shall assume that some such function is also regularly exempli­fied in normal scientific practice and can be isolated in writings whose pur­pose is not exclusively pedagogic. At this point we need only notice that on the question of practice, textbooks provide no evidence whatsoever. No text­book ever included a table tha t either intended or managed to infirm the the­ory the text was written to describe. Readers of current science texts accept the theories there expounded on the authority of the author and the scientific community, not because of any tables tha t these texts contain. If the tables are read at all, as they often are, they are read for another reason.

I shall inquire for this other reason in a moment but must first remark on the second putative function of measurement, tha t of exploration. Numerical data like those collected in the right-hand column of our table can, it is often supposed, be useful in suggesting new scientific theories or laws. Some people seem to take for granted tha t numerical data are more likely to be productive of new generalizations than any other sort. It is that special pro­ductivity, rather than measurement’s function in confirmation, that probably accounts for Kelvin’s dictum’s being inscribed on the facade a t the University of Chicago.6

I t is by no means obvious th a t our ideas about this function of numbers are related to the textbook schema outlined in the diagram above, yet I see no other way to account for the special efficacy often attributed to the results of measurement. We are, I suspect, here confronted with a vestige of an ad­mittedly outworn belief that laws and theories can be arrived a t by some

8 P rofessor F rank Knight, for example, sug- ‘I f you cannot m easure, m easure anyhow.’ ” gcsts tha t to social scientists the “practical Eleven T w e n ty -S ix , p. 169. m eaning [of Kelvin’s statem ent) tends to be:

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process like “ running the machine backwards.” Given the numerical data in the “Experiment” column of the table, logico-mathematical manipulation (aided, all would now insist, by “ intuition” ) can proceed to the statem ent of the laws that underlie the numbers. If any process even remotely like this is involved in discovery— if, tha t is, laws and theories are forged directly from data by the mind—then the superiority of numerical to qualitative data is immediately apparent. The results of measurement are neutral and precise; they cannot mislead. Even more important, numbers are subject to mathe­matical manipulation; more than any other form of data, they can be assimi­lated to the semimechanical textbook schema.

I have already implied my skepticism about these two prevalent descrip­tions of the function of measurement. In Sections I I and I I I each of these functions will be further compared with ordinary scientific practice. But it will help first critically to pursue our examination of textbook tables. By doing so I would hope to suggest that our stereotypes about measurement do not even quite fit the textbook schema from which they seem to derive. Though the numerical tables in a textbook do not there function either for exploration or confirmation, they are there for a reason. T hat reason we may perhaps discover by asking what the author of a text can mean when he says that the numbers in the “Theory” and “Experiment” column of a table “agree.”

At best the criterion must be in agreement within the limits of accuracy of the measuring instruments employed. Since computation from theory can usually be pushed to any desired number of decimal places, exact or numeri­cal agreement is impossible in principle. But anyone who has examined the tables in which the results of theory and experiment are compared must rec­ognize tha t agreement of this more modest sort is rather rare. Almost always the application of a physical theory involves some approximation (in fact, the plane is not “ frictionless,” the vacuum is not “perfect,” the atoms are not “unaffected” by collisions), and the theory is not therefore expected to yield quite precise results. Or the construction of the instrument may involve ap­proximations (e.g., the “ linearity” of vacuum tube characteristics) that cast doubt upon the significance of the last decimal place that can be unambigu­ously read from their dial. Or it may simply be recognized that, for reasons not clearly understood, the theory whose results have been tabulated or the instrument used in measurement provides only estimates. For one of these reasons or another, physical scientists rarely expect agreement quite within instrumental limits. In fact, they often distrust it when they see it. At least on a student lab report overly close agreement is usually taken as presum p­tive evidence of data manipulation. T hat no experiment gives quite the ex­pected numerical result is sometimes called “The F ifth Law of Therm ody­namics.”® The fact that, unlike some other scientific laws, it has acknowledged exceptions does not diminish its utility as a guiding principle.

I t follows tha t what scientists seek in numerical tables is not usually “agree-

® T he first three Law s of Therm odynam ics paratus w orks the first tim e it is set up. W e a rc well known outside the trade. T he “F ourth shall exam ine evidence for the F ifth Law Law " states tha t no piece of experim ental ap- below.

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m ent” at all, but what they often call “ reasonable agreement.” Furtherm ore, if we now ask for a criterion of “ reasonable agreement,” we are literally forced to look in the tables themselves. Scientific practice exhibits no consistently applied or consistently applicable external criterion. “ Reasonable agreement” varies from one part of science to another, and within any part of science it varies with time. W hat to Ptolemy and his immediate successors was reason­able agreement between astronomical theory and observation was to Coper­nicus incisive evidence that the Ptolemaic system must be wrong.7 Between the times of Cavendish (1731-1810) and Ramsay (1852-1916), a similar change in accepted chemical criteria for “ reasonable agreement” led to the study of the noble gases.8 These divergences are typical and they are matched by those between contemporary branches of the scientific community. In parts of spectroscopy “reasonable agreement” means agreement in the first six or eight left-hand digits in the numbers of a table of wave lengths. In the theory of solids, by contrast, two-place agreement is often considered very good in­deed. Yet there are parts of astronomy in which any search for even so lim­ited an agreement m ust seem utopian. In the theoretical study of stellar mag­nitudes agreement to a multiplicative factor of ten is often taken to be “ reasonable.”

Notice that we have now inadvertently answered the question from which we began. We have, tha t is, said what “agreement” between theory and ex­periment must mean if tha t criterion is to be drawn from the tables of a science text. But in doing so we have gone full circle. I began by asking, at least by implication, what characteristic the numbers of the table must ex­hibit if they are to be said to “agree.” I now conclude that the only possible criterion is the mere fact that they appear, together with the theory from which they arc derived, in a professionally accepted text. When they appear in a text, tables of numbers drawn from theory and experiments cannot dem­onstrate anything but “ reasonable agreement.” And even that they demon­strate only by tautology, since they alone provide the definition of “ reasonable agreement” that has been accepted by the profession. T hat, I think, is why the tables are there: they define “ reasonable agreement.” By studying them, the reader learns what can be expected of the theory. An acquaintance with the tables is part of an acquaintance with the theory itself. W ithout the tables, the theory would be essentially incomplete. W ith respect to measure­ment, it would be not so much untested as untestable. Which brings us very close to the conclusion tha t, once it has been embodied in a text— which for present purposes means, once it has been adopted by the profession—no the­ory is recognized to be testable by any quantitative tests that it has not al­ready passed.9

7 T . S. Kuhn, T h e Copem ican Revolution the distinction between analytic and synthetic(Cam bridge, Mass., 1957), pp. 72-76, 135-143. tru th . T o the ex ten t tha t a scientific theory

8 W illiam Ram say, T he Gases o f the A t - must be accompanied by a statem ent of thetnosphcre: the H isto ry o f T heir D iscovery evidence for it in o rder to have em pirical (London, 1896), Chapters 4 and 5. meaning, the full theory (w hich includes the

* T o pursue th is point would ca rry us far evidence) m ust be analytically true. F o r abeyond the subject of th is paper, but it should statem ent of the philosophical problem of ana-bc pursued because, if I am righ t, it relates to lyticity sec W . V. Quine, “T w o D ogm as of the im portant contem porary controversy over Em piricism ” and other essays in F rom a L ogi-

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Perhaps these conclusions are not surprising. Certainly they should not be. Textbooks are, after all, written some time after the discoveries and confirma­tion procedures whose outcomes they record. Furtherm ore, they are written for purposes of pedagogy. The objective of a textbook is to provide the reader, in the most economical and easily assimilable form, with a statem ent of what the contemporary scientific community believes it knows and of the principal uses to which that knowledge can be put. Information about the ways in which that knowledge was acquired (discovery) and in which it was enforced on the profession (confirmation) would at best be excess baggage. Though including tha t information would almost certainly increase the “ humanistic” values of the text and might conceivably breed more flexible and creative scientists, it would inevitably detract from the ease of learning the contemporary scien­tific language. To date only the last objective has been taken seriously by most writers of textbooks in the natural sciences. As a result, though texts may be the right place for philosophers to discover the logical structure of finished scientific theories, they are more likely to mislead than to help the unwary individual who asks about productive methods. One might equally appropriately go to a college language text for an authoritative characteriza­tion of the corresponding literature. Language texts, like science texts, teach how to read literature, not how to create or evaluate it. W hat signposts they supply to these latter points are most likely to point in the wrong direction.10

II. M OTIVES FOR NORM AL M EA SU REM EN T

These considerations dictate our next step. We must ask how measurement comes to be juxtaposed with laws and theories in science texts. Furthermore, we must go for an answer to the journal literature, the medium through which natural scientists report their own original work and in which they evaluate that done by others.” Recourse to this body of literature immediately casts doubt upon one implication of the standard textbook schema. Only a minis­cule fraction of even the best and most creative measurements undertaken by

cal Point o f V iezv (Cam bridge, M ass., 1953). cation whose origins a t least can be found inF o r a stim ulating, but loose, discussion of the the seventeenth century and which has in-occasionally analytic sta tus of sdcntific laws, creased in rigor ever since. T here w as a timesee N. R . H anson, P atterns o f D iscovery (different in different sciences) when the pat-(C am bridge, England, 1958), pp. 93-118. A tern of communication in a science was muchnew discussion of the philosophical problem, the same as th a t still visible in the hum anitiesincluding copious references to the controver- and m any of the social sciences, but in all thesial literature, is Alan Pasch, Experience and physical sciences th is pattern is a t least a cen-the A n a ly tic : A Reconsideration o f E m piri- tu ry gone, and in many of them it disappearedctsnt (Chicago, 1958). even ea rlie r than th a t Now all publication of

10 T he m onograph cited in note 3 will argue research results occurs in journals read onlyth a t the m isdirection supplied by science tex ts by the profession. Rooks are exclusively tcx t-is both system atic and functional. It is by no books, compendia, popularizations, o r plulo-mcans clear that a m ore accurate im age of the sophical reflections, and w riting them is ascientific processes would enhance the research somewhat suspect, because nonprofessional, ac-efliciency of physical scientists. tivity. Needless to say th is sharp and rigid

11 It is, of course, som ewhat anachronistic s e r r a t io n between articles and books, researchto apply the term s “journal lite ra tu re’' and and nonrcsearch w ritings, g reatly increases“textbooks" in the whole of the period I have the strength of w hat I have callcd the tex t-been asked to discuss. But I am concerned to book image.emphasize a pattern of professional communi-

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natural scientists are motivated by a desire to discover new quantitative regu­larities or to confirm old ones. Almost as small a fraction turn out to have had either of these effects. There are a few that did so, and I shall have something to say about them in Sections I I I and IV. But it will help first to discover just why these exploratory and confirmatory measurements are so rare. In this section and most of the next, I therefore restrict myself to meas­urement’s most usual function in the normal practice of science.12

Probably the rarest and most profound sort of genius in physical science is that displayed by men who, like Newton, Lavoisier, or Einstein, enunciate a whole new theory that brings potential order to a vast number of natural phenomena. Yet radical reformulations of this sort are extremely rare, largely because the state of science very seldom provides occasion for them. M ore­over, they are not the only truly essential and creative events in the develop­ment of scientific knowledge. The new order provided by a revolutionary new theory in the natural sciences is always overwhelmingly a potential order. Much work and skill, together with occasional genius, are required to make it actual. And actual it must be made, for only through the process of actu­alization can occasions for new theoretical reformulations be discovered. The bulk of scientific practice is thus a complex and consuming mopping-up op­eration tha t consolidates the ground made available by the most recent theo­retical breakthrough and tha t provides essential preparation for the break­through to follow. In such mopping-up operations, measurement has its overwhelmingly most common scientific function.

Ju st how im portant and difficult these consolidating operations can be is indicated by the present state of Einstein’s general theory of relativity. The equations embodying that theory have proved so difficult to apply tha t (ex­cluding the limiting case in which the equations reduce to those of special relativity) they have so far yielded only three predictions tha t can be com­pared with observation.18 M en of undoubted genius have totally failed to develop others, and the problem remains worth their attention. Until it is solved, Einstein’s general theory remains a largely fruitless, because unexploit- able, achievement.14

Undoubtedly the general theory of relativity is an extreme case, but the situation it illustrates is typical. Consider, for a somewhat more extended example, the problem that engaged much of the best eighteenth-century scien­

12 H ere and elsewhere in th is paper I ignore perihelion of M ercury , and the red shift ofthe very large am ount of measurem ent done light from d istant sta rs. Only the first tw osim ply to gather factual inform ation. I think are actually quantitative predictions in theof such m easurem ents as specific gravities, present sta te of the theory,wave lengths, spring constants, boiling points, 14 T he difficulties in producing concrete ap- ctc„ undertaken in o rder to determ ine param e- plications of the general theory of relativ ityters that must be inserted into scientific Uieo- need not prevent scientists from attem pting tories but whose numerical outcome those thco- exploit the scicntific view point embodied inries do not (o r did not in the relevant period) tha t theory. But, perhaps unfortunately, itpredict. T his sort of measurem ent is not w ith- seems to be doing so. U nlike the special thc- out interest, hut I think it w idely understood, ory, general relativity is today very little stud-In any case, considering it would too g rea tly ied by students of physics. W ith in fifty yearsextend the limits of this paper. we may conceivably have to tally lost sight of

13These a re : the deflection of light in the th is aspect of E inste in 's contribution,sun’s gravitational field, the procession of the

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tific thought, tha t of deriving testable numerical predictions from Newton’s three Laws of motion and from his principle of universal gravitation. When N ewton’s theory was first enunciated late in the seventeenth century, only his Third Law (equality of action and reaction) could be directly investigated by experiment, and the relevant experiments applied only to very special cases.15 The first direct and unequivocal demonstrations of the Second Law awaited the development of the Atwood machine, a subtly conceived piece of labora­tory apparatus that was not invented until almost a century after the appear­ance of the Principia.*• Direct quantitative investigations of gravitational attraction proved even more difficult and were not presented in the scientific literature until 1798.11 Newton’s F irst Law cannot, to this day, be directly compared with the results of laboratory measurement, though developments in rocketry make it likely that we have not much longer to wait.

I t is, of course, direct demonstrations, like those of Atwood, that figure most largely in natural science texts and in elementary laboratory exercises. Be­cause simple and unequivocal, they have the greatest pedagogic value. T hat they were not and could scarcely have been available for more than a century after the publication of Newton’s work makes no pedagogic difference. At most it only leads us to mistake the nature of scientific achievement.1** But if Newton’s contemporaries and successors had been forced to wait tha t long for quantitative evidence, apparatus capable of providing it would never have been designed. Fortunately there was another route, and much eigthteenth- century scientific talent followed it. Complex mathematical manipulations, exploiting all the laws together, perm itted a few other sorts of prediction cap­able of being compared with quantitative observation, particularly with lab­oratory observations of pendula and with astronomical observations of the motions of the moon and planets. But these predictions presented another and equally severe problem, that of essential approximations.1* The suspen-

15 T he most relevant and widely employed o f Cavendish’s measurem ents o f 1798, but it isexperim ents were perform ed with pendula. De- only a fte r Cavendish that measurem ent beginsterm ination of the recoil when tw o pendulum to yield consistent results.bobs collided seems to have been the main ** M odem laboratory apparatus designed toconceptual and experim ental tool used in the help students study Galileo’s law of free fallseventeenth century to determ ine w hat dy- provides a classic, though perhaps quite neces-namical "action '’ and "reaction” were. See A. sary, exam ple of the way pedagogy misdirectsW olf, A H istory o f Science, Technology, and the historical im agination about the relation bc-Philosophy in the 16th & 17th Centuries, new tween creative science and m easurem ent. Nonecd. prepared by D. M cKic (London, 1950), of the apparatus now used could possibly havepp. 155, 231-235: and R. Dugas, La m ican iqu t been built in the seventeenth century. O ne ofau xvii* siecle (N euchatel, 1954), pp. 283-298; the best and most widely disseminated piecesand S ir Isaac N n v to n 's M athetnatical Princi- of equipment, fo r example, allows a heavy bobpies o f N atural P hilosophy and h is Sys tem o f to fall between a pair of parallel vertical rails.the W orld, cd. F . C ajori (B erkeley, 1934), pp. These rails a rc electrically charged every 21-28. W olf (p. 155) describes the T h ird Law l/100 th of a second, and the spark tha t thenas "the only physical law of the three.’’ passes through the bob from rail to rail re-

16 See the excellent description of th is ap- cords the bob’s position on a chemically trea t-paratus and the discussion of A tw ood’s reasons cd tape. O ther picccs of apparatus involvefor building it in H anson, Patterns o f D is- electric tim ers, etc. F o r the h istorical difiicul-coi'ery, pp. 100-102 ami notes to these pages. ties of m aking m easurem ents relevant to this

17 A. W olf, A H isto ry o f Science, Technol- law, see below.ogy, and Philosophy in the E ighteenth Cen- 19 A ll the applications of N ew ton’s Lawstury, 2nd cd. revised by D. M cK ic (London, involve approxim ations of some so rt, but in1952), pp. 111-113. T here a re some precursors the following exam ples the approxim ations

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sions of laboratory pendula are neither weightless nor perfectly elastic; air resistance damps the motion of the bob; besides, the bob itself is of finite size, and there is the question of which point of the bob should be used in comput­ing the pendulum’s length. If these three aspects of the experimental situation are neglected, only the roughest sort of quantitative agreement between theory and observation can be expected. But determining how to reduce them (only the last is fully eliminable) and what allowance to make for the residue are themselves problems of the utmost difficulty. Since Newton’s day much bril­liant research has been devoted to their challenge.20

The problems encountered when applying Newton’s Laws to astronomical prediction are even more revealing. Since each of the bodies in the solar sys­tem attracts and is attracted by every other, precise prediction of celestial phenomena demanded, in Newton’s day, the application of his Laws to the simultaneous motions and interactions of eight celestial bodies. (These were the sun, moon, and six known planets. I ignore the other planetary satellites.) The result is a mathematical problem that has never been solved exactly. To get equations tha t could be solved, Newton was forced to the simplifying as­sumption that each of the planets was attracted only by the sun, and the moon only by the earth. W ith this assumption, he was able to derive K epler’s famous Laws, a wonderfully convincing argument for his theory. But deviation of planets from the motions predicted by Kepler’s Laws is quite apparent to simple quantitative telescopic observation. To discover how to treat these deviations by Newtonian theory, it was necessary to devise mathematical estimates of the “perturbations” produced in a basically Keplerian orbit by the interplanetary forces neglected in the initial derivation of Kepler’s Laws. Newton’s mathematical genius was displayed at its best when he produced a first crude estimate for the perturbation of the moon’s motion caused by the sun. Improving his answer and developing similar approximate answers for the planets exercised the greatest mathematical minds of the eighteenth and early nineteenth centuries, including those of Euler, Lagrange, Laplace, and Gauss.21 Only as a result of their work was it possible to recognize the anom­aly in M ercury’s motion tha t was ultimately to be explained by Einstein’s general theory. T hat anomaly had previously been hidden within the limits of “ reasonable agreement.”

As far as it goes, the situation illustrated by quantitative application of Newton’s Laws is, I think perfectly typical. Similar examples could be pro­duced from the history of the corpuscular, the wave, or the quantum mechan­ical theory of light, from the history of electromagnetic theory, quantitative chemical analysis, or any other of the numerous natural scientific theories with quantitative implications. In each of these cases, it proved immensely difficult to find many problems that permitted quantitative comparison of theory and observation. Even when such problems were found, the highest scientific tal­ents were often required to invent apparatus, reduce perturbing effects, and

liave a quantitative im portance that they do work.not possess in those tha t precede. 21 Ibid., pp. 96-101. W illiam W hew ell, H is-

20 W olf, Eighteenth Century, pp. 75-81, pro- tn ry o f the Inductive Sciences, rev. ed., 3 vols. vides a good prelim inary description of this (London, 1847), II, 213-271.

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estimate the allowance to be made for those tha t remained. This is the sort of work that most physical scientists do most of the time insofar as their work is quantitative. Its objective is, on the one hand, to improve the measure of “ reasonable agreement” characteristic of the theory in a given application and, on the other, to open up new areas of application and establish new measures of “ reasonable agreement” applicable to them. For anyone who finds mathe­matical or manipulative puzzles challenging, this can be fascinating and in­tensely rewarding work. And there is always the remote possibility tha t it will pay an additional dividend: something may go wrong.

Yet unless something does go wrong— a situation to be explored in Section IV— these finer and finer investigations of the quantitative match between theory and observation cannot be described as attem pts a t discovery or a t confirmation. The man who is successful proves his talents, but he does so by getting a result tha t the entire scientific community had anticipated some­one would someday achieve. His success lies only in the explicit demonstra­tion of a previously implicit agreement between theory and the world. No novelty in nature has been revealed. Nor can the scientist who is successful in this sort of work quite be said to have “confirmed” the theory tha t guided his research. For if success in his venture “confirms” the theory, then failure ought certainly “ infirm” it, and nothing of the sort is true in this case. Fail­ure to solve one of these puzzles counts only against the scientist; he has put in a great deal of time on a project whose outcome is not worth publication; the conclusion to be drawn, if any, is only th a t his talents were not adequate to it. I f measurement ever leads to discovery or to confirmation, it does not do so in the most usual of all its applications.

I I I . T H E EFFEC T S OF NORM AL M EA SU REM EN TThere is a second significant aspect of the normal problem of measurement

in natural science. So far we have considered why scientists usually measure; nowr we must consider the results tha t they get when they do so. Imm ed­iately another stereotype enforced by textbooks is called in question. In textbooks the numbers tha t result from measurement usually appear as the archetypes of the “ irreducible and stubborn facts” to which the scientist must, by struggle, make his theories conform. But in scientific practice, as seen through the journal literature, the scientist often seems rather to be strug­gling with facts, trying to force them into conformity with a theory he does not doubt. Quantitative facts cease to seem simply “ the given.” They must be fought for and with, and in this fight the theory with which they are to be compared proves the most potent weapon. Often scientists cannot get numbers that compare well with theory until they know what numbers they should be making nature yield.

P art of this problem is simply the difficulty in finding techniques and instru­ments that permit the comparison of theory with quantitative measurements. We have already seen tha t it took almost a century to invent a machine that could give a straightforward quantitative demonstration of Newton’s Second Law. Hut the machine that Charles Atwood described in 1784 was not the first instrument to yield quantitative information relevant to that Law.

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Attempts in this direction had been made ever since Galileo’s description of his classic inclined plane experiment in 1638.28 Galileo’s brilliant intuition had seen in this laboratory device a way of investigating how a body moves when acted upon only by its own weight. After the experiment he announced that measurement of the distance covered in a measured time by a sphere rolling down the plane confirmed his prior thesis tha t the motion was uniformly accelerated. As reinterpreted by Newton, this result exemplified the Second Law for the special case of a uniform force. But Galileo did not report the numbers he had gotten, and a group of the best scientists in France announced their total failure to get comparable results. In p rin t they wondered whether Galileo could himself have tried the experiment.”

In fact, it is almost certain tha t Galileo did perform the experiment. If he did, he must surely have gotten quantitative results tha t seemed to him in adequate agreement with the law ( s = V& at2) tha t he had shown to be a con­sequence of uniform acceleration. But anyone who has noted the stop-watches or electric timers, and the long planes or heavy llyw’heels needed to perform this experiment in modern elementary laboratories may legitimately suspect tha t Galileo’s results were not in unequivocal agreement with his law. Quite possibly the French group looking even a t the same data would have wondered how they could seem to exemplify uniform acceleration. This is, of course, largely speculation. But the speculative element casts no doubt upon my present point: whatever its source, disagreement between Galileo and those who tried to repeat his experiment was entirely natural. If Galileo’s generali­zation had not sent men to the very border of existing instrumentation, an area in which experimental scatter and disagreement about interpretation were inevitable, then no genius would have been required to make it. His example typifies one im portant aspect of theoretical genius in the natural sciences— it is a genius that leaps ahead of the facts, leaving the rather different talent of the experimentalist and instrumentalist to catch up. In this case catching up took a long time. The Atwood Machine was designed because, in the middle of the eighteenth century, some of the best Continental scientists still wondered whether acceleration provided the proper measure of force. Though their doubts derived from more than measurement, measurement was still sufficiently equivocal to fit a variety of different quantitative conclusions.**

The preceding example illustrates the difficulties and displays the role of theory in reducing scatter in the results of measurement. There is, however, more to the problem. When measurement is insecure, one of the tests for reliability of existing instruments and manipulative techniques must inevitably be their ability to give results that compare favorably with existing theory. In some parts of natural science, the adequacy of experimental technique can be judged only in this way. W hen that occurs, one may not even speak of “ in­secure” instrumentation or technique, implying th a t these could be improved without recourse to an external theoretical standard.

22 F or a modern English version of the o r- 23 T his whole sto ry and m ore is brilliantly iginal sec Galileo Galilei, Dialogues Concern- set forth in A. K o y ri, "A n Experim ent in m g T w o N e w Sciences, trans. H enry Crew M easurem ent,” Proc. A m cr. Phil. Soc., 1953, and A. De Salvio (E vanston and Chicago, 97: 222-237.1946), pp. 171-172. 21 H anson, Patterns o f D iscovery, p. 101.

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For example, when John Dalton first conceived of using chemical measure­ments to elaborate an atomic theory tha t he had initially drawn from meteoro­logical and physical observations, he began by searching the existing chemical literature for relevant data. Soon he realized that significant illumination could be obtained from those groups of reactions in which a single pair of elements, e.g., nitrogen and oxygen, entered into more than one chemical com­bination. If his atomic theory were right, the constituent molecules of these compounds should differ only in the ratio of the number of whole atoms of each element tha t they contained. The three oxides of nitrogen might, for example, have molecules N 20 , NO, and NO*, or they might have some other similarly simple arrangem ent.25 But whatever the particular arrangem ents, if the weight of nitrogen were the same in the samples of the three oxides, then the weights of oxygen in the three samples should be related to each other by simple whole-number proportions. Generalization of this principle to all groups of compounds formed from the same group of elements produced D alton’s Law of M ultiple Proportions.

Needless to say, D alton’s search of the literature yielded some data that, in his view, sufficiently supported the Law. But— and this is the point of the illustration— much of the then extant data did not support D alton’s Law a t all. For example, the measurements of the French chemist Proust on the two oxides of copper yielded, for a given weight of copper, a weight ratio for oxygen of 1.47:1. On D alton’s theory the ratio ought to have been 2:1, and Proust is just the chemist who might have been expected to confirm the prediction. He was, in the first place, a fine experimentalist. Besides, he was then engaged in a major controversy involving the oxides of copper, a controversy in which he upheld a view very close to D alton’s. But, at the beginning of the nineteenth century, chemists did not know how to perform quantitative analyses that displayed multiple proportions. By 1850 they had learned, but only by letting D alton’s theory lead them. Knowing what results they should expect from chemical analyses, chemists were able to devise techniques th a t got them. As a result chemistry texts can now state tha t quantitative analysis confirms Dalton’s atomism and forget that, historically, the relevant analytic techniques are based upon the very theory they are said to confirm. Before D alton’s theory was announced, measurement did not give the same results. There are self-fulfilling prophecies in the physical as well as in the social sciences.

T hat example seems to me quite typical of the way measurement responds to theory in many parts of the natural sciences. I am less sure that my next, and far stranger, example is equally typical, but colleagues in nuclear physics assure me that they repeatedly encounter similar irreversible shifts in the results of measurement.

25 T h is is not, of course, D alton’s original L. K. N ash, T he A tom ic M olecular Theory.notation. In fact, I am somewhat m odernizing H arv ard Case H istories in Experim ental Sci-and simplifying th is whole account. It can be ence, Case 4 (C am bridge, Mass., 1950") ; andreconstructed more fully fro m : A. N . M el- “T he O rigins of D alton’s Chemical A tomicdrum , “T he Development of the A tom ic T he- Theory ,” Isis , 1956, 47: 110-116. See also theo ry : (1 ) Berthollet’s D octrine of V ariab le useful discussions of atom ic w eight scatteredProportions,” M anch. M em ., 1910, 54: 1-16; th rough J . R . Parting ton , A Short H isto ry ofand “ (6 ) T he Reception accorded to the T he- C hem istry, 2nd ed. (1-ondon, 1951).ory advocated by D alton.” ibid., 1911, 55: 1-10;

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Very early in the nineteenth century, P. S. de Laplace, perhaps the greatest and certainly the most famous physicist of his day, suggested that the recently observed heating of a gas when rapidly compressed might explain one of the outstanding numerical discrepancies of theoretical physics. This was the dis­agreement, approximately 20 per cent, between the predicted and measured values of the speed of sound in air— a discrepancy that had attracted the a t­tention of all Europe’s best mathematical physicists since Newton had first pointed it out. When Laplace’s suggestion was made, it defied numerical con­firmation (note the recurrence of this typical difficulty), because it demanded refined measurements of the thermal properties of gases, measurements that were beyond the capacity of apparatus designed for measurements on solids and liquids. But the French Academy offered a prize for such measurements, and in 1819 the prize was won by two brilliant young experimentalists, Delaroche and Berard, men whose names arc still cited in contemporary scientific litera­ture. Laplace immediately made use of these measurements in an indirect theoretical computation of the speed of sound in air, and the discrepancy between theory and measurement dropped from 20 per cent to 2.5 per cent, a recognized triumph in view of the state of measurement.2®

But today no one can explain how' this triumph can have occurred. Laplace’s interpretation of Delaroche and B crard’s figures made use of the caloric theory in a region where our own science is quite certain th a t tha t theory differs from directly relevant quantitative experiment by about 40 per cent. There is, how­ever, also a 12 per cent discrepancy between the measurements of Delaroche and Berard and the results of equivalent experiments today. We are no longer able to get their quantitative result. Yet, in Laplace’s perfectly straightforward and essential computation from the theory, these two discrepancies, experi­mental and theoretical, cancelled to give close final agreement between the pre­dicted and measured speed of sound. We may not, I feel sure, dismiss this as the result of mere sloppiness. Both the theoretician and the experimentalists involved were men of the very highest caliber. Rather we must here see evi­dence of the way in which theory and experiment may guide each other in the exploration of areas new to both.

These examples may enforce the point drawn initially from the examples in the last section. Exploring the agreement between theory and experiment into new areas or to new’ limits of precision is a difficult, unremitting, and, for many, exciting job. Though its object is neither discovery nor confirmation, its appeal is quite sufficient to consume almost the entire time and attention of those physical scientists who do quantitative work. I t demands the very best of their imagination, intuition, and vigilance. In addition—when combined w-ith those of the last section—these examples may show something more. They may, that is, indicate why new laws of nature are so very seldom discovered simply by inspecting the results of measurements made without advance knowledge of those laws. Because most scientific laws have so few quantitative points of contact with nature, because investigations of those contact points usually demand such laborious instrumentation and approximation, and because nature itself needs to be forced to yield the appropriate results, the route from theory

20 T . S. Kuhn. “T he Caloric T heory of A diabatic Com pression,” Isis. 1958, 49: 132-140.

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or law to measurement can almost never be travelled backwards. Numbers gathered without some knowledge of the regularity to be expected almost never speak for themselves. Almost certainly they remain just numbers.

This does not mean that no one has ever discovered a quantitative regularity merely by measuring. Boyle’s Law relating gas pressure with gas volume, Hooke’s Law relating spring distortion with applied force, and Joule’s relation­ship between heat generated, electrical resistance, and electric current were all the direct results of measurement. There are other examples besides. But, partly just because they are so exceptional and partly because they never occur until the scientist measuring knows everything but the particular form of the quantitative result he will obtain, these exceptions show just how improbable quantitative discovery by quantitative measurement is. T he cases of Galileo and Dalton— men who intuited a quantitative result as the simplest expression of a qualitative conclusion and then fought nature to confirm it—are very much the more typical scientific events. In fact, even Boyle did not find his Law until both he and two of his readers had suggested that precisely that law (the simplest quantitative form that yielded the observed qualitative regu­larity) ought to result if the numerical results were recorded.21 Here, too, the quantitative implications of a qualitative theory led the way.

One more example may make clear a t least some of the prerequisites for this exceptional sort of discovery. The experimental search for a law or laws describing the variation with distance of the forces between magnetized and between electrically charged bodies began in the seventeenth century and was actively pursued through the eighteenth. Yet only in the decades immediately preceding Coulomb’s classic investigations of 1785 did measurement yield even an approximately unequivocal answer to these questions. W hat made the dif­ference between success and failure seems to have been the belated assimila­tion of a lesson learned from a part of Newtonian theory. Simple force laws, like the inverse square law for gravitational attraction, can generally be ex­pected only between mathematical points or bodies tha t approximate to them. The more complex laws of attraction between gross bodies can be derived from the simpler law governing the attraction of points by summing all the forces between all the pairs of points in the two bodies. But these laws will seldom take a simple mathematical form unless the distance between the two bodies is large compared with the dimensions of the attracting bodies them ­selves. Under these circumstances the bodies will behave as points, and ex­periment may reveal the resulting simple regularity.

Consider only the historically simpler case of electrical attractions and re­pulsions.28 During the first half of the eighteenth century— when electrical forces were explained as the results of effluvia emitted by the entire charged body— almost every experimental investigation of the force law involved plac­ing a charged body a measured distance below one pan of a balance and then

27 M arie Boas, R obert Doyle and S even - E lectricity jrotr. the G reeks to Coulomb, H a r-teenth-Century C hem istry (Cam bridge, E ng- vard Case H istories in Experim ental Science,land, 1958), p. 44. Case 8 (C am bridge, M ass., 1954), and in

28 M uch relevant m aterial w ill be found in W olf, Eighteenth C entury, pp. 239-250, 268-D uane Roller and D uane H . D. Roller, T h e D e- 271.v e h e m e n t o f the Concept o f E lectric Charge:

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measuring the weight tha t had to be placed in the other pan to just overcome the attraction. W ith this arrangem ent of apparatus, the attraction varies in no simple way with distance. Furtherm ore, the complex way in which it does vary depends critically upon the size and material of the attracted pan. M any of the men who tried this technique therefore concluded by throwing up their hands; others suggested a variety of laws including both the inverse square and the inverse first power; measurement had proved totally equivocal. Yet it did not have to be so. W hat was needed and what was gradually acquired from more qualitative investigations during the middle decades of the century was a more ‘‘Newtonian” approach to the analysis of electrical and magnetic phenomena.2" As this evolved, experimentalists increasingly sought not the attraction between bodies but that between point poles and point charges. In tha t form the experimental problem was rapidly and unequivocally resolved.

This illustration shows once again how large an amount of theory is needed before the results of measurement can be expected to make sense. But, and this is perhaps the main point, when tha t much theory is available, the law is very likely to have been guessed without measurement. Coulomb’s result, in particular, seems to have surprised few scientists. Though his measure­ments were necessary to produce a firm consensus about electrical and mag­netic attractions— they had to be done; science cannot survive on guesses - many practitioners had already concluded that the law of attraction and re­pulsion must be inverse square. Some had done so by simple analagy to New­ton’s gravitational law; others by a more elaborate theoretical argument; still others from equivocal data. Coulomb’s Law was very much “ in the a ir” before its discoverer turned to the problem. If it had not been, Coulomb might not have been able to make nature yield it.

[Repeated discussions of this Section indicate two respects in which my text may be misleading. Some readers take my argument to mean that the com­mitted scientist can make nature yield any measurements tha t he pleases. A few of these readers, and some others as well, also think my paper asserts that for the development of science, experiment is of decidedly secondary impor­tance when compared with theory. Undoubtedly the fault is mine, but I intend to be making neither of these points.

If what I have said is right, nature undoubtedly responds to the theoretical predispositions with which she is approached by the measuring scientist. But that is not to say either that nature will respond to any theory at all or that she will ever respond very much. Reexamine, for a historically typical exam­ple, the relationship between the caloric and dynamical theory of heat. In their abstract structures and in the conceptual entities they presuppose, these two theories are quite different and, in fact, incompatible. But, during the years

t0 A fuller account would have to describe ton 's Principia. F o r the differences between both the earlier and the la ter approaches as these books, their influence in the eighteenth “ New tonian.” T he conception that electric century, and their impact on the development force results from eflluvia is partly C artesian of electrical theory, see I. 13. Cohen, Franklin but in the eighteenth century its locus-clossicus and N civ tor.: A n Inquiry into Speculative was the aether theory developed in N ew ton’s N r a t onion E xperim enta l Science and F rank - O ptieks. Coulomb’s approach and tha t of sev- I in’s IVork in E lectricity as an E xam ple Thcre- eral of his contem poraries depends far m ore o f (Philadelphia, 1956). directly on the m athem atical theory in New-

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when the two vied for the allegiance of the scientific community, the theoreti­cal predictions that could be derived from them were very nearly the same (see the reference cited in note 26). If they had not been, the caloric theory would never have been a widely accepted tool of professional research nor would it have succeeded in disclosing the very problems th a t made transition to the dynamical theory possible. I t follows tha t any measurement which, like that of Delaroche and Berard, “ fit” one of these theories must have “ very nearly fit” the other, and it is only within the experimental spread covered by the phrase “ very nearly” that nature proved able to respond to the theoretical predisposition of the measurer.

T hat response could not have occurred with “any theory a t all.” There are logically possible theories of, say, heat tha t no sane scientist could ever have made nature fit, and there are problems, mostly philosophical, tha t make it worth inventing and examining theories of tha t sort. But those are not our problems, because those merely “conceivable” theories are not among the op­tions open to the practicing scientist. His concern is with theories that seem to fit what is known about nature, and all these theories, however different their structure, will necessarily seem to yield very similar predictive results. If they can be distinguished a t all by measurements, those measurements will usually strain the limits of existing experimental techniques. Furtherm ore, within the limits imposed by those techniques, the numerical differences at issue will very often prove to be quite small. Only under these conditions and within these limits can one expect nature to respond to preconception. On the other hand, these conditions and limits arc just the ones typical in the his­torical situation.

If this much about my approach is clear, the second possible misunder­standing can be dealt with more easily. By insisting tha t a quite highly de­veloped body of theory is ordinarily prerequisite to fruitful measurement in the physical sciences, I may seem to have implied that in these sciences theory must always lead experiment and tha t the latter has a t best a decidedly sec­ondary role. But tha t implication depends upon identifying “experiment” with “measurement,” an identification I have already explicitly disavowed. It is only because significant quantitative comparison of theories with nature comes a t such a late stage in the development of a science that theory has seemed to have so decisive a lead. If we had been discussing the qualitative experimentation tha t dominates the earlier developmental stages of a physical science and th a t continues to play a role later on, the balance would be quite different. Perhaps, even then, we would not wish to say tha t experiment is prior to theory (though experience surely is), but we would certainly find vastly more symmetry and continuity in the ongoing dialogue between the two. Only some of my conclusions about the role of measurement in physical sci­ence can be readily extrapolated to experimentation a t large.]

IV. EX TRA O R D IN A R Y M EA SU REM EN TTo this point I have restricted attention to the role of measurement in the

normal practice of natural science, the sort of practice in which all scientists are mostly, and most scientists are always, engaged. But natural science also

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displays abnormal situations— times when research projects go consistently astray and when no usual techniques seem quite to restore them—and it is through these rare situations that measurement shows its greatest strengths. In particular, it is through abnormal states of scientific research tha t measure­ment comes occasionally to play a major role in discovery and in confirmation.

Let me first try to clarify what I mean by an “abnormal situation” or by what I am elsewhere calling a “crisis state.”30 I have already indicated that it is a response by some part of the scientific community to its awareness of an anomaly in the ordinarily concordant relationship between theory and ex­periment. But it is not, let us be clear, a response called forth by any and every anomaly. As the preceding pages have shown, current scientific practice always embraces countless discrepancies between theory and experiment. D ur­ing the course of his career, every natural scientist again and again notices and passes by qualitative and quantitative anomalies th a t just conceivably might, if pursued, have resulted in fundamental discovery. Isolated discrep­ancies with this potential occur so regularly tha t no scientist could bring his research problems to a conclusion if he paused for many of them. In any case, experience has repeatedly shown that, in overwhelming proportion, these dis­crepancies disappear upon closer scrutiny. They may prove to be instrumen­tal effects, or they may result from previously unnoticed approximations in the theory, or they may, simply and mysteriously, cease to occur when the experiment is repeated under slightly different conditions. M ore often than not the efficient procedure is therefore to decide tha t the problem has “gone sour,” that it presents hidden complexities, and tha t it is time to put it aside in favor of another. Fortunately or not, tha t is good scientific procedure.

But anomalies are not always dismissed, and of course they should not be. I f the effect is particularly large when compared with well-established meas­ures of “ reasonable agreement” applicable to similar problems, or if it seems to resemble other difficulties encountered repeatedly before, or if, for personal reasons, it intrigues the experimenter, then a special research project is likely to be dedicated to it.*1 At tha t point the discrepancy will probably vanish through an adjustm ent of theory or apparatus; as we have seen, few anoma­lies resist persistent effort for long. But it may resist, and. if it docs, we may have the beginning of a “crisis” or “abnormal situation” affecting those in whose usual area of research the continuing discrepancy lies. They, at least, having exhausted all the usual recourses of approximation and instrum enta­tion, may be forced to recognize tha t something has gone wrong, and their behavior as scientists will change accordingly. At this point, to a vastly greater extent than a t any other, the scientist will s ta rt to search at random, trying anything at all which he thinks may conceivably illuminate the nature of his difficulty. If tha t difficulty endures long enough, he and his colleagues may even begin to wonder whether their entire approach to the now problematic range of natural phenomena is not somehow askew.

30 See note 3. "T h e Case of the F loppy-E ared R abbits: An31 A recent exam ple of the factor? deter- Instance o f Serendipity Gained and Serendipity

mining pursuit of an anom aly has been investi- Lost,” A m cr. Soc. R ev ., 1958, 64: 128-136. gated by Bernard Barber and Renee C. Fox,

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This is, of course, an immensely condensed and schematic description. U n­fortunately, it will have to remain so, for the anatomy of the crisis state in natural science is beyond the scope of this paper. I shall remark only that these crises vary greatly in scope: they may emerge and be resolved within the work of an individual; more often they will involve most of those engaged in a particular scientific specialty; occasionally they will engross most of the members of an entire scientific profession. But, however widespread their im­pact, there arc only a few ways in which they may be resolved. Sometimes, as has often happened in chemistry and astronomy, more refined experimental techniques or a finer scrutiny of the theoretical approximations will eliminate the discrepancy entirely. On other occasions, though I think not often, a dis­crepancy that has repeatedly defied analysis is simply left as a known anom­aly, encysted within the body of more successful applications of the theory. Newton’s theoretical value for the speed of sound and the observed preces­sion of M ercury’s perihelion provide obvious examples of effects which, though since explained, remained in the scientific literature as known anomalies for half a century or more. But there are still other modes of resolution, and it is they which give crises in science their fundamental importance. Often crises are resolved by the discover)' of a new natural phenomenon; occasionally their resolution demands a fundamental revision of existing theory.

Obviously crisis is not a prerequisite for discovery in the natural sciences. We have already noticed that some discoveries, like that of Boyle’s Law and of Coulomb’s Law, emerge naturally as a quantitative specification of what is qualitatively already known. M any other discoveries, more often qualitative than quantitative, result from preliminary exploration with a new instrument, e.g., the telescope, battery , or cyclotron. In addition, there arc the famous “accidental discoveries,” Galvani and the twitching frog’s legs, Roentgen and X-rays, Becquerel and the fogged photographic plates. T he last two categories of discovery arc not, however, always independent of crises. I t is probably the ability to recognize a significant anomaly against the background of cur­rent theory tha t most distinguishes the successful victim of an “accident” from those of his contemporaries who passed the same phenomenon by. (Is this not part of the sense of Pasteur’s famous phrase, “ In the fields of observation, chance favors only the prepared mind” ?)*2 In addition, the new instrumental techniques that multiply discoveries are often themselves by-products of crises. Volta’s invention of the battery wras, for example, the outcome of a long a t­tempt to assimilate Galvani’s observations of frogs’ legs to existing electrical theory. And, over and above these somewhat questionable cases, there are a large number of discoveries tha t arc quite clearly the outcome of prior crises The discovery of the planet Neptune was the product of an effort to account for known anomalies in the orbit of Uranus.35 The nature of both chlorine and carbon monoxide was discovered through attem pts to reconcile Lavoisier’s new chemistry with observation.54 The so-called noble gases were the prod-

32 From P asteu r’s inaugural address a t Lille 51 F or chlorine see E rn s t von M eyer. Ain 1854 as quoted in Rene V allery-R adot, La H isto ry o f C hem istry from the Earliest T unes V ie de P asteur (P a ris , 1903), p. 88. to the Present D ay, trans. G. M ’Gowan (Lon-

83 A ngus A rm itage, A C entury o f A stron - don. 1891), pp. 224-227. F o r carbon monoxide o m y (London, 1950), pp. 111-115. see J . R . Parting ton , A S h o r t H isto ry o f

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ucts of a long series of investigations initiated by a small bu t persistent anom­aly in the measured density of nitrogen.54 The electron was posited to explain some anomalous properties of electrical conduction through gases, and its spin was suggested to account for other sorts of anomalies observed in atomic spec­tra.'1'1 T he discovery of the neutrino presents still another example, and the list could be extended.37

I am not certain how large these discoveries-through-anomaly would rank in a statistical survey of discovery in the natural sciences.3* They are, how­ever, certainly important, and they require disproportionate emphasis in this paper. To the extent that measurement and quantitative technique play an especially significant role in scientific discovery, they do so precisely because, by displaying serious anomaly, they tell scientists when and where to look for a new qualitative phenomenon. T o the nature of tha t phenomenon, they usu­ally provide no clues. When measurement departs from theory, it is likely to yield mere numbers, and their very neutrality makes them particularly sterile as a source of remedial suggestions. But numbers register the departure from theory with an authority and finesse tha t no qualitative technique can dupli­cate, and tha t departure is often enough to s tart a search. N eptune might, like Uranus, have been discovered through an accidental observation; it had, in fact, been noticed by a few earlier observers who had taken it for a pre­viously unobserved star. W hat was needed to draw attention to it and to make its discovery as nearly inevitable as historical events can be was its involve­ment, as a source of trouble, in existing quantitative observation and existing theory. I t is hard to see how either electron-spin or the neutrino could have been discovered in any other way.

The case both for crises and for measurement becomes vastly stronger as soon as we turn from the discovery of new natural phenomena to the inven­tion of fundamental new theories. Though the sources of individual theoreti­cal inspiration may be inscrutable (certainly they must remain so for this pa}>er), the conditions under which inspiration occurs is not. 1 know of no fundamental theoretical innovation in natural science whose enunciation has not been preceded by clear recognition, often common to most of the profes­sion, that something was the m atter with the theory then in vogue. T he state

Chem istry, 2nd cd. (London, 1948), pp. 140- and N uclear P hysics (N ew Y ork, 1958), pp.141; and J. R. P arting ton and D. M cKic, 328-330. I know of no o ther elem entary ac-"H istorical S tudies of the Phlogiston T h eo ry : count recent enough to include a descriptionIV . L ast Phases of the T heory ,” A nnals o f of the physical detection of the neutrino.Science, 1939, 4: 365. Because scientific attention is often con-

3S See note 7. centrated upon problems th a t sccni to displayw F o r useful surveys of the experim ents anom aly, the prevalence of discovery-through-

which led to the discovery of the electron sec anom aly may be one reason for the prevalenceT . W . Chalmers, H istoric Researches: Chap- of sim ultaneous discovery in the sciences. F orters in the H isto ry o f P hysical and Chem ical evidence tha t it is not the only one see T . S.D iscovery (London, 1949), pp. 187-217, and K uhn, “Conservation of E nergy as an Exam -J. J . Thom son, Recollections and Reflections pie of Sim ultaneous Discovery,” Critical Prob-(N ew York, 1937), pp. 325-371. F o r electron- lem s in the H isto ry o f Science, ed. M arshallspin sec F. K. Richtm eycr, E. H . K cnnard , C lagctt (M adison, 1959), pp. 321-356, but no-aud T . Lauritscn, Introduction to M odern tice tha t much of w hat is there said about thePhysics, 5th ed. (N ew Y ork, 1955), p. 212. em ergence of “conversion processes'’ also dc-

87 R ogers D. Rusk, Introduction to A tom ic scribes the evolution of a crisis state.

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of Ptolemaic astronomy was a scandal before Copernicus’ announcement.30 Both Galileo’s and Newton’s contributions to the study of motion were ini­tially focused upon difficulties discovered in ancient and medieval theory.40 Newton’s new theory of light and color originated in the discovery tha t exist­ing theory would not account for the length of the spectrum, and the wave theory tha t replaced Newton’s was announced in the midst of growing concern about anomalies in the relation of diffraction and polarization to Newton’s theory.41 Lavoisier’s new chemistry was born after the observation of anoma­lous weight relations in combustion; thermodynamics from the collision of two existing nineteenth-century physical theories; quantum mechanics from a variety of difficulties surrounding black-body radiation, specific heat, and the photoelectric effect.42 Furtherm ore, though this is not the place to show it, each of these difficulties, except the optical one observed by Newton, had been a source of concern before (but usually not long before) the theory that re­solved it was announced.

I suggest, therefore, that though a crisis or an ‘‘abnormal situation” is only

39 Kuhn, Copem ican Revolution, pp. 138- 140. 270-271; A. R. H all. T h e Scientific R evo ­lution. 1S00-1S00 (London, 1954), pp. 13-17. N ote particu larly the role of agitation for calendar reform in intensifying the crisis.

40 Kuhn, Copem ican Revolution , pp. 237- 260, and item s in bibliography on pp. 290-291.

41 F o r N ew ton see T . S. Kuhn, "N ew ton’s Optical Papers," in Isaac N eiv ton's Papers & L etters on N a tura l Philosophy, cd. I. B. Cohen (C am bridge, Mass., 1958), pp. 27-45. F or the wave theory sec E . T . W hittaker, H isto ry o f the Theories o f A ether and E lectricity , The Classical Theories, 2nd cd. (I^ondon, 1951), pp. 94-109, and W hcw cll, Inductive Sciences, I I , 396-466. These references clearly delineate the crisis that characterized optics when Fres- ncl independently began to develop the wave theory afte r 1812. But they say too little about eighteenth-century developments to indicate a crisis p rior to Young’s earlier defense of the wave theory in and a fte r 1801. In fact, it is not a ltogether clear t in t there w as one, o r at least tliat there was a new one. N ew ton’s co r­puscular theory of light had never been quite universally acccpted, and Y oung's early oppo­sition to it w as based entirely upon anom alies tliat liad been generally recognized and often exploited before. W e may need to conclude tha t most of the eighteenth century w as ch ar­acterized by a low-level crisis in optics, for the dominant theory w as never immune to fundamental criticism and attack .

T h a t would be sufficient to m ake the point that is of concern here, but I suspect a careful Study of the eighteenth-century optical lite ra ­tu re will perm it a still stronger conclusion. A cursory look a t tha t body of litera tu re sug­gests tha t the anom alies of Newtonian optics w ere fa r m ore apparent and pressing in the tw o decades before Y oung's w ork than they

had been before. D uring the 1780’s the availa­bility of achrom atic lenses and prism s led to num erous proposals for an astronom ical deter­mination of the relative motion of the sun and stars. (T h e references in W hittaker, op. cit., p. 109, lead directly to a fa r la rger litera tu re .) But these all depended upon ligh t's moving ntore quickly in glass than in a ir and thus gave new relevance to an old controversy. L’A bbi H auv dem onstrated experim entally (M em . de I’Acad. (1788), pp. 34-60) tha t H uy- ghcn’s w ave-theoretical treatm ent of double refraction had yielded better results than N ew ­ton’s corpuscular treatm ent. T h e resulting problem leads to the prize offered by the F rench Academy in 1808 and thus to M alus’ discovery of polarization by rcflcction in the same year. O r again, the Philosophical T rans­actions for 1796, 1797, and 1798 contain a series of tw o articles by Brougltam and a third by P rcvost which show still o ther difficulties in N ew ton’s theory. According to P rcvost, in particular, the sorts of forces which must be exerted on light a t an interface in order to explain reflection and refraction arc not com ­patible w ith the sorts of forces needed to ex ­plain inflection (P h il. Trans., 1798, 84: 325- 328. B iographers of Young m ight pay m ore attention than they have to the tw o Brougham papers in the preceding volumes. These dis­play an intellectual com mitm ent th a t goes a long way to explain Brougham 's subsequent vitriolic attack upon Young in the pages of the Edinburgh Review.)

4* R ichtm eyer el a!., M odem Physics, pp. 89-94, 124-132, and 409-414. A m ore ele­m entary account of the black-body problem and of the photoelectric effect is included in G erald H olton, Introduction to Concepts and Theories in Physical Science (Cam bridge, Mass., 1953), pp. 528-545.

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one of the routes to discovery in the natural sciences, it is prerequisite to fundamental inventions of theory. Furtherm ore, I suspect th a t in the crea­tion of the particularly deep crisis tha t usually precedes theoretical innova­tion, measurement makes one of its two most significant contributions to scientific advance. M ost of the anomalies isolated in the preceding paragraph were quantitative or had a significant quantitative component, and, though the subject again carries us beyond the bounds of this essay, there is excellent reason why this should have been the case.

Unlike discoveries of new natural phenomena, innovations in scientific the­ory are not simply additions to the sum of what is already known. Almost always (always, in the m ature sciences) the acceptance of a new theory de­mands the rejection of an older one. In the realm of theory, innovation is thus necessarily destructive as well as constructive. But, as the preceding pages have repeatedly indicated, theories are, even more than laboratory in­struments, the essential tools of the scientist’s trade. W ithout their constant assistance, even the observations and measurements made by the scientist would scarcely be scientific. A threat to theory is therefore a threat to the scientific life, and, though the scientific enterprise progresses through such threats, the individual scientist ignores them while he can. Particularly, he ignores them if his own prior practice has already committed him to the use of the threatened theory.44 I t follows that new theoretical suggestions, de­structive of old practices, rarely if ever emerge in the absence of a crisis that can no longer be suppressed.

No crisis is, however, so hard to suppress as one that derives from a quanti­tative anomaly that has resisted all the usual efforts a t reconciliation. Once the relevant measurements have been stabilized and the theoretical approxi­mations fully investigated, a quantitative discrepancy proves persistently ob­trusive to a degree that few qualitative anomalies can match. By their very nature, qualitative anomalies usually suggest ad hoc modifications of theory tha t will disguise them, and once these modifications have been suggested there is little way of telling whether they are “good enough.” An established quantitative anomaly, in contrast, usually suggests nothing except trouble, but a t its best it provides a razor-sharp instrument for judging the adequacy of proposed solutions. Kepler provides a brilliant case in point. After prolonged struggle to rid astronomy of pronounced quantitative anomalies in the motion of M ars, he invented a theory accurate to 8' of arc, a measure of agreement tha t would have astounded and delighted any astronomer who did not have access to the brilliant observations of Tycho Brahe. But from long experience Kepler knew Brahe's observations to be accurate to 4 ' of arc. To us, he said, Divine goodness has given a most diligent observer in Tycho Brahe, and it is therefore right that we should with a grateful mind make use of this gift to find the true celestial motions. Kepler next attem pted computations with non­

43 Evidence for this cffcct of prio r experi- the la tte r phenomenon needs no citation. A nence w ith a theory is provided by the w ell- earlier and particularly m oving version o f theknown, but inadequately investigated, youth- same sentim ent is provided by D arw in in thefulness o f fam ous innovators as well a s by the last chapter of T h e Ori(/in o f Species. (Secway in which younger men tend to cluster to the 6th ed. [N ew York, 1889J, I I , 295-296.)the new er theory. P lanck’s statem ent about

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circular figures. The outcome of those trials was his first two Laws of plane­tary motion, the Laws th a t for the first time made the Copernican system work.4*

Two brief examples should make clear the differential effectiveness of quali­tative and quantitative anomalies. Newton was apparently led to his new the­ory of light and color by observing the surprising elongation of the solar spectrum. Opponents of his new theory quickly pointed out that the exist­ence of elongation had been known before and th a t it could be treated by existing theory. Qualitatively they were quite right. But utilizing Snell’s quantitative law of refraction (a law that had been available to scientists for less than three decades), Newton was able to show that the elongation pre­dicted by existing theory was quantitatively far smaller than the one observed. On this quantitative discrepancy, all previous qualitative explanations of elon­gation broke down. Given the quantitative law of refraction, Newton’s ulti­mate, and in this case quite rapid, victory was assured.45 The development of chemistry provides a second striking illustration. I t was well known, long be­fore Lavoisier, tha t some metals gain weight when they arc calcined (i.e., roasted). Furtherm ore, by the middle of the eighteenth century this qualita­tive observation was recognized to be incompatible with at least the simplest versions of the phlogiston theory, a theory tha t said phlogiston escaped from the metal during calcination. But so long as the discrepancy remained quali­tative, it could be disposed of in numerous ways: perhaps phlogiston had negative weight, or perhaps fire particles lodged in the roasted metal. There were other suggestions besides, and together they served to reduce the urgency of the qualitative problem. T he development of pneumatic techniques, how­ever, transformed the qualitative anomaly into a quantitative one. In the hands of Lavoisier, they showed how much weight was gained and where it came from. These were data with which the earlier qualitative theories could not deal. Though phlogiston’s adherents gave vehement and skillful battle, and though their qualitative arguments were fairly persuasive, the quantita­tive arguments for Lavoisier’s theory proved overwhelming.4®

These examples were introduced to illustrate how difficult it is to explain away established quantitative anomalies, and to show how much more effec­tive these are than qualitative anomalies in establishing uncvadablc scientific crises. But the examples also show something more. They indicate that meas­urement can be an immensely powerful weapon in the battle between two

44 J . L. E . D rcycr, A H isto ry o f A stronom y peri mental Scicncc, Case 2 (C am bridge, Mass.,from Thales to Kepler, 2nd cd. (N ew York, 1950), and D . M cKic, A nto ine iMVoisicr:1953), pp. 385-393. S cien tist, Econom ist, Social R eform er (N ew

46 K uhn, “ N ew ton’s O ptical Papers,” pp. Y ork, 1952). M aurice Daum as, Lavoisier,31-36. Theoricien e t exp trim en tcur (P a ris , 1955) is

46 T his is a slight oversim plification, since the best recent scholarly review. J . H . W hite,the Iwttle between I-avoisier's new chem istry T h e PMogisto>v T heory (London, 1932) andand its opponents really im plicated m ore tl»an especially J . R . P arting ton and D. M cKie,combustion processes, and the full range of "H isto rica l S tudies of the Phlogiston T h eo ry :relevant evidence cannot be treated in term s of IV . L ast Phases of the T heory ,” A nnals ofcombustion alone. Useful elem entary accounts Science, 1939, 4: 113-149, give most detailof Lavoisier’s contributions can be found in : about the conflict between the new theory andJ . B. Conant, T he O verthrow o f the Phlogis- the old.ton T heory, H arvard Case H istories in E x -

MEASUREMENT IN MODERN PHYSICAL SCIENCE 183

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theories, and that, I think, is its second particularly significant function. F ur­thermore, it is for this function—aid in the choice between theories— and for it alone, that we must reserve the word “confirmation.” We must, th a t is, if “confirmation” is intended to denote a procedure anything like what scientists ever do. The measurements that display an anomaly and thus create crisis may tempt the scientist to leave science or to transfer his attention to some other part of the field. But, if he stays where he is, anomalous observations, quantitative or qualitative, cannot tem pt him to abandon his theory until an­other one is suggested to replace it. Just as a carpenter, while he retains his craft, cannot discard his toolbox because it contains no hammer fit to drive a particular nail, so the practitioner of science cannot discard established the­ory because of a felt inadequacy. At least he cannot do so until shown some other way to do his job. In scientific practice the real confirmation questions always involve the comparison of two theories with each other and with the world, not the comparison of a single theory with the world. In these three- way comparisons, measurement has a particular advantage.

To sec where measurement’s advantage resides, I must once more step briefly, and hence dogmatically, beyond the bounds of this essay. In the tran ­sition from an earlier to a later theory, there is very often a loss as well as a gain of explanatory power.47 Newton’s theory of planetary and projectile mo­tion was fought vehemently for more than a generation because, unlike its main competitors, it demanded the introduction of an inexplicable force that acted directly upon bodies a t a distance. Cartesian theory, for example, had attem pted to explain gravity in terms of the direct collisions between elemen­tary particles. T o accept Newton meant to abandon the possibility of any such explanation, or so it seemed to most of Newton’s immediate successors.48 Similarly, though the historical detail is more equivocal, Lavoisier’s new chemi­cal theory was opposed by a number of men who felt that it deprived chem­istry of one principal traditional function—the explanation of the qualitative properties of bodies in term s of the particular combination of chemical “prin­ciples” that composed them.40 In each case the new theory was victorious, but the price of victory was the abandonment of an old and partly achieved goal. For eighteentli-century Newtonians it gradually became “ unscientific” to ask for the cause of gravity; nineteenth-century chemists increasingly ceased to ask for the causes of particular qualities. Yet subsequent experience has shown that there was nothing intrinsically “ unscientific” about these questions. Gen­

*1 T h is point is ccntral to the reference ticularly pp. 331-336. M uch essential m aterialcitcd in note 3. In fact, it is largely the ncccs- is also scattered th rough H €fenc M etzger, L essity of balancing gains and losses and the con- .rw'i® <i la fin dtt xviii* sicete, vol. I (P aris ,trovcrsies th a t so often result from disagree- 1923). and NeitiOH, S tah l. Poerltaave, et lam ents about an appropriate balancc tha t m ake doctrine chimiquc (P a ris . 1930). N otice par-it appropriate to describe changes of theory as ticularly tha t the phlogistonists. w ho looked“revolutions.0 upon o res as elem entary bodies from which

48 Cohen, Franklin and N ew ton , C hapter 4 ; the m etals w ere compounded by addition ofP ierre B runet, [ .’introduction des theories de phlogiston, could explain w hy the m etals wereN ew ton en France au . m ? siecle (P a ris , so much m ore like each o ther than w ere the1931). ores from w hich they w ere compounded. All

49 On this traditional task of chem istry sec m etals had a principle, phlogiston, in common.E. Meyer.son. Identity and R eality, trans. K. N o such explanation w as possible on Lavoi-Low cnberg (London, 1930), Chapter X , par- sier’s theory.

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eral relativity does explain gravitational attraction, and quantum mechanics does explain many of the qualitative characteristics of bodies. We now know what makes some bodies yellow and others transparent, etc. But in gaining this immensely im portant understanding, we have had to regress, in certain respects, to an older set of notions about the bounds of scientific inquiry. Problems and solutions that had to be abandoned in embracing classic theo­ries of modern science are again very much with us.

The study of the confirmation procedures as they are practiced in the sci­ences is therefore often the study of what scientists will and will not give up in order to gain other particular advantages. T hat problem has scarcely even been stated before, and I can therefore scarcely guess what its fuller investi­gation would reveal. But impressionistic study strongly suggests one signifi­cant conclusion. I know of no case in the development of science which exhibits a loss of quantitative accuracy as a consequence of the transition from an earlier to a later theory. Nor can I imagine a debate between scien­tists in which, however hot the emotions, the search for greater numerical accuracy in a previously quantified field would be called “ unscientific.” Prob­ably for the same reasons tha t make them particularly effective in creating scientific crises, the comparison of numerical predictions, where they have been available, has proved particularly successful in bringing scientific con­troversies to a close. W hatever the price in redefinitions of science, its meth­ods, and its goals, scientists have shown themselves consistently unwilling to compromise the numerical success of their theories. Presumably there are other such desiderata as well, but one suspects that, in case of conflict, meas­urement would be the consistent victor.

V. M EA SU R EM EN T IN T H E D EV ELO PM EN T OF PHYSICAL SC IEN C E

T o this point we have taken for granted th a t measurement did play a cen­tral role in physical science and have asked about the nature of that role and the reasons for its peculiar efficacy. Now we must ask, though too late to anticipate a comparably full response, about the way in which physical sci­ence came to make use of quantitative techniques at all. To make tha t large and factual question manageable, I select for discussion only those parts of an answer which relate particularly closely to what has already been said.

One recurrent implication of the preceding discussion is that much qualita­tive research, both empirical and thcorctical, is normally prerequisite to fruit­ful quantification of a given research field. In the absence of such prior work, the methodological directive, “ Go ye forth and measure,” may well prove only an invitation to waste time. I f doubts about this point remain, they should be quickly resolved by a brief review of the role played by quantitative tech­niques in the emergence of the various physical sciences. Let me begin by asking what role such techniques had in the scientific revolution that centered in the seventeenth century.

Since any answer must now be schematic, I begin by dividing the fields of physical science studied during the seventeenth century into two groups. The

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first, to be labeled the traditional sciences, consists of astronomy, optics, and mechanics, all of them fields th a t had received considerable qualitative and quantitative development in antiquity and during the Middle Ages. These fields are to be contrasted with what I shall call the Baconian sciences, a new cluster of research areas tha t owed their status as scicticcs to the seventeenth century’s characteristic insistence upon experimentation and upon the com­pilation of natural histories, including histories of the crafts. To this second group belong particularly the study of heat, of electricity, of magnetism, and of chemistry. Only chemistry had been much explored before the Scientific Revolution, and the men who explored it had almost all been either craftsmen or alchemists. If we except a few of the a r t’s Islamic practitioners, the emer­gence of a rational and systematic chemical tradition cannot be dated earlier than the late sixteenth century.30 M agnetism, heat, and electricity emerged still more slowly as independent subjects for learned study. Even more clearly than chemistry, they are novel by-products of the Baconian elements in the “new philosophy.”M

The separation of traditional from Baconian sciences provides an important analytic tool, because the man who looks to the Scientific Revolution for ex­amples of productive measurement in physical science will find them only in the sciences of the first group. Further, and perhaps more revealing, even in these traditional sciences measurement was most often effective ju s t when it could be performed with well-known instruments and applied to very nearly traditional concepts. In astronomy, for example, it was Tycho Brahe’s en­larged and better-calibrated version of medieval instruments tha t made the decisive quantitative contribution. The telescope, a characteristic novelty of the seventeenth century, was scarcely used quantitatively until the last third of the century, and that quantitative use had no effect on astronomical theory until Bradley’s discovery of aberration in 1729. Even that discovery was iso­lated. Only during the second half of the eighteenth century did astronomy begin to experience the full effects of the immense improvements in quanti­tative observation that the telescope perm itted.” Or again, as previously in­dicated, the novel inclined plane experiments of the seventeenth century were not nearly accurate enough to have alone been the source of the law of uni­form acceleration. W hat is im portant about them—and they are critically im portant— is the conception tha t such measurements could have relevance to the problems of free fall and of projectile motion. T hat conception implies a fundamental shift in both the idea of motion and the techniques relevant to its analysis. But clearly no such conception could have evolved as it did if

60 Boas, R obrrt Hoyle, pp. 48-66. equally satisfactory discussion of the dcvclop-51 F o r clcctricity see, R oller and R oller, m cnt of therm al science before the eighteenth

Concept o f E lectric Charge. H arvard Case ccntury. but W olf, 16th and 17th Centuries.H isto ries in Experim ental Science, Case 8 pp. 82-92 and 275-281 will illustrate the trans-(C am bridpc, Mass.. 1954), and. E dgar Zilscl, form ation produced by Baconianism."T h e O rigins of W illiam G ilbert's Scicntific 52 W olf, Eighteenth Century, pp. 102-145,M ethod,” J. H ist. Ideas. 1941, 2: 1-32. I agree and W hcw ell, Inductive Sciences. pp. 213-371.w ith those who feel Zilscl exaggerates the im- P articu larly in the latter, notice the difficultyportance of a single factor in the genesis of in separating advances due to improved in-clcctrical science and. by implication, o f Bacon- strum cntation from those due to improvedianism, but the craft influences he describes theory. T h is difficulty is not due prim arily tocannot conceivably be dismissed. T h ere is no W hcw ell's mode of presentation.

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many of the subsidiary concepts needed for its exploitation had not existed, a t least as developed embryos, in the works of Archimedes and of the scho­lastic analysts of motion.53 Here again the effectiveness of quantitative work depended upon a long-standing prior tradition.

Perhaps the best test case is provided by optics, the third of my traditional sciences. In this field during the seventeenth century, real quantitative work was done with both new and old instruments, and the work done with old instruments on well-known phenomena proved the more important. The Sci­entific Revolution’s reformulation of optical theory turned upon Newton’s prism experiments, and for these there was much qualitative precedent. New­ton’s innovation was the quantitative analysis of a well-known qualitative ef­fect, and that analysis was possibly only because of the discovery, a few decades before Newton’s work, of Snell’s law of refraction. T h a t law is the vital quantitative novelty in the optics of the seventeenth century. I t was, however, a law that had been sought by a scries of brilliant investigators since the time of Ptolemy, and all had used apparatus quite similar to that which Snell employed. In short, the research which led to Newton’s new theory of light and color was of an essentially traditional nature.44

Much in seventeenth-century optics was, however, by no means traditional. Interference, diffraction, and double refraction were all first discovered in the half-century before Newton’s Opticks appeared; all were totally unexpected phenomena; and all were known to Newton.” On two of them Newton con­ducted careful quantitative investigations. Yet the real impact of these novel phenomena upon optical theory was scarcely felt until the work of Young and Fresnel a century later. Though Newton was able to develop a brilliant pre­liminary theory for interference effects, neither he nor his immediate succes­sors even noted that that theory agreed with quantitative experiment only for the limited case of perpendicular incidence. Newton’s measurements of dif­fraction produced only the most qualitative theory, and on double refraction he seems not even to have attem pted quantitative work of his own. Both Newton and Huyghen announced mathematical laws governing the refraction of the extraordinary ray, and the latter showed how to account for this be­havior by considering the expansion of a spheroidal wave front. But both mathematical discussions involved large extrapolations from scattered quanti­tative data of doubtful accuracy. And almost a hundred years elapsed before quantitative experiments proved able to distinguish between these two quite different mathematical formulations.5* As with the other optical phenomena

a* F o r pre-Galilean w ork sec, M arshall see the rcfercncc in (he preceding: note. TheClagett, T he Science o f M echanics in the cightccnth-ccntury investigations of theseM iddle A g es (M adison, W is., 1959), particu- phenomena have scarcely been studied, but forlarly P a rts II & I I I . F o r Galileo’s use of this w hat is known see, Joseph P riestley, H istorywork see, A lexandre Koyrc. t.tudes Calile- . . . o f D iscoveries relating to V ision. L ight,ennes, 3 vols. (P a ris , 1939), particularly I and Colours (London. 1772), pp. 279-316, 498-& II. 520, 548-562. T he earliest examples I know of

f * A. C. Crombie, A ugustine to Galileo m ore precise w ork on double refraction are,(London, 1952), pp. 70-82, and W olf, 16th & R . J . H atiy , "S u r la double refraction du17th Centuries, pp. 244-254. Spath d ’Islandc,” M em . d I'Acad. (1788), pp.

56 Ibid., pp. 254-264. 34-61, and, W . H . W ollaston, "O n the obliqueM F or the seventeenth-century w ork (in- R efraction of Iceland C rystal,” Phil. Trans.,

eluding H uyghen’s geom etric construction) 1802, 92: 381-386.

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discovered during the Scientific Revolution, most of the eighteenth century was needed for the additional exploration and instrumentation prerequisite to quantitative exploitation.

Turning now to the Baconian sciences, which throughout the Scientific Revolution possessed few old instrum ents and even fewer well-wrought con­cepts, we find quantification proceeding even more slowly. Though the seven­teenth century saw many new instrum ents, of which a number were quanti­tative and others potentially so, only the new barometer disclosed significant quantitative regularities when applied to new fields of study. And even the barometer is only an apparent exception, for pneumatics, the field of its ap­plication, was able to borrow en bloc the concepts of a far older field, hydro­statics. As Toricelli put it, the barometer measured pressure “at the bottom of an ocean of the clement air.”57 In the field of magnetism the only signifi­cant seventeenth-century measurements, those of declination and dip, were made with one or another modified version of the traditional compass, and these measurements did little to improve the understanding of magnetic phe­nomena. For a more fundamental quantification, magnetism, like electricity, awaited the work of Coulomb, Gauss, Poisson, and others in the late eight­eenth and early nineteenth centuries. Before that work could be done, a better qualitative understanding of attraction, repulsion, conduction, and other such phenomena was needed. The instruments which produced a lasting quantifi­cation had then to be designed with these initially qualitative conceptions in mind.5* Furthermore, the decades in which success was a t last achieved are almost the same ones that produced the first effective contacts between meas­urement and theory in the study of chemistry and of heat.59 Successful quan­tification of the Baconian sciences had scarcely begun before the last third of the eighteenth century and only realized its full potential in the nineteenth. T hat realization—exemplified in the work of Fourier, Clausius, Kelvin, and Maxwell— is one facet of a second scientific revolution no less consequential than the seventeenth-century revolution. Only in the nineteenth century did the Baconian physical sciences undergo the transformation which the group of traditional sciences had experienced two or more centuries before.

Since Professor Guerlac’s paper is devoted to chemistry and sincc I have al­ready sketched some of the bars to quantification of electrical and magnetic phenomena, I take my single more extended illustration from the study of heat. Unfortunately, much of the research upon which such a sketch should be based remains to be done. W hat follows is necessarily more tentative than what has gone before.

M any of the early experiments involving thermometers read like investi­gations of that new instrum ent rather than like investigations w ith it. How

67 See I.H .B . and A .G .H . Spiers, T h e P h ys- 53-66; and W . C. W alker, “T he Detection andical Treatises o f Pascal (N ew Y ork, 1937), Estim ation of E lectric C harge in the E ight-p. 164. T h is whole volume displays the w ay eenth C entury,” A nna ls o f Science, 1936, 1:in which seventeenth-century pneum atics took 66-100.over concepts from hydrostatics. 69 F o r heat sec, D ouglas M cK ie and N. II.

6S F or the quantification and early m athe- de V . H cathcotc, T he D iscovery o f Specificm atization of electrical science, see: R oller and L aten t H ea ts (London, 1935). In chem-and Roller, Concept o f E lectric Charge, pp. is try it may well be impossible to fix any date66-80; W hittaker, A eth er and E lectricity, pp. for the “first effective contacts between mcas-

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could anything else have been the case during a period when it was totally unclear what the thermometer measured? Its readings obviously depended upon the “degree of heat” but apparently in immensely complex ways. “D e­gree of heat” had for a long time been defined by the senses, and the senses responded quite differently to bodies which produced the same thermometric readings. Before the thermometer could become unequivocally a laboratory instrum ent rather than an experimental subject, thermometric reading had to be seen as the direct measure of “ degree of heat,” and sensation had simul­taneously to be viewed as a complex and equivocal phenomenon dependent upon a number of different parameters.**

T hat conceptual reorientation seems to have been completed in a t least a few scientific circles before the end of the seventeenth century, but no rapid discovery of quantitative regularities followed. F irst scientists had to be forced to a bifurcation of “degree of heat” into “quantity of heat,” on the one hand, and “ tem perature,” on the other. In addition they had to select for close scrutiny, from the immense multitude of available thermal phenomena, the ones that could most readily be made to reveal quantitative law. These proved to be: mixing two components of a single fluid initially a t different temperatures, and radiant heating of two different fluids in identical vessels. Even when attention was focused upon these phenomena, however, scientists still did not get unequivocal or uniform results. As Heathcote and McKie have brilliantly shown, the last stages in the development of the concepts of specific and latent heat display intuited hypotheses constantly interacting with stubborn measurement, each forcing the other into line.*1 Still other sorts of work were required before the contributions of Laplace, Poisson, and Fourier could transform the study of thermal phenomena into a branch of m athe­matical physics.82

This sort of pattern, reiterated both in the other Baconian sciences and in the extension of traditional sciences to new instruments and new phenomena, thus provides one additional illustration of this paper’s most persistent thesis. The road from scientific law to scientific measurement can rarely be traveledurcm ent and theory.” V olum etric o r g rav i- account of the slow stages in the deploymentm ctric m easures w ere always an ingredient of of the therm om eter as a scicntific instrum ent,chemical recipes and assays. By the seven- Robert Boyle's N e w E xperim ents and Obser-tccnth century, fo r exam ple in the w ork of vations Touching Cold illustrates the sevcn-Boyle, w eight-gain o r loss was often a clue tccnth century’s need to dem onstrate tha t prop-to the thcorctical analysis of particu lar re- crly constructed therm om eters m ust rcplaccactions. But until the middle of the eighteenth the senses in therm al m easurem ents evencentury, the significance of chemical measure- though the tw o give divergent results. Secment seems alw ays to have been either dc- W orks o f the Honourable Robert B oyle, cd.scriptive (a s in recipes) or qualitative (a s in T . Birch, 5 vols. (London, 1744), II, 240-243.dem onstrating a w ciglu-gain w ithout signifi- 61 F o r the elaboration of calorim ctric con- cant reference to its m agnitude). Only in the ccpts sec, E . M ach, D ie Principien der W iir-w ork of Black, Lavoisier, and R ichter docs m elehre (Leipzig, 1919), pp. 153-181, and Mc-measurcm ent begin to play a fully quantitative Kie and H eathcote, Specific and Latent H eats.role in the development of chemical laws and T he discussion of K rafft’s w ork in the la ttertheories. F or an introduction to these men and (pp. 59-63) provides a particularly strikingtheir w ork see, J . B. P arting ton , A S h o r t H is- exam ple of the problems in m aking measurc-to ry o f C hem istry, 2nd cd. (London, 1951), ment work.pp. 93-97, 122-128, and 161-163. ** Gaston B achelard, E tude sur Involution

00 M aurice D aum as, L es instrum ents scien- d ’un probleme de physique (P a ris , 1928), andtifiques a u x xvii* e t xviii* sitc les (P a ris , Kuhn, "C aloric T heory of A diabatic Com prcs-1953), pp. 78-80, provides an excellent b rief sion.”

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in the reverse direction. To discover quantitative regularity one must nor­mally know what regularity one is seeking and one’s instruments must be designed accordingly; even then nature may not yield consistent or general- izable results without a struggle. So much for my major thesis. The preceding remarks about the way in which quantification entered the modern physical sciences should, however, also recall this paper’s minor thesis, for they re­direct attention to the immense efficacy of quantitative experimentation un­dertaken within the context of a fully mathematized theory. Sometime be­tween 1800 and 1850 there was an im portant change in the character of research in many of the physical sciences, particularly in the cluster of re­search fields known as physics. T h a t changc is what makes me call the mathe- matization of Baconian physical science one facet of a second scientific revolution.

I t would be absurd to pretend tha t mathematization was more than a facet. The first half of the nineteenth century also witnessed a vast increase in the scale of the scientific enterprise, major changes in patterns of scientific or­ganization, and a total reconstruction of scientific education.63 But these changes affected all the sciences in much the same way. They ought not to explain the characteristics that differentiate the newly mathematized sciences of the nineteenth century from other sciences of the same period. Though my sources are now impressionistic, I feel quite sure that there are such char- asteristics. Let me hazard the following prediction. Analytic, and in part statistical, research would show that physicists, as a group, have displayed since about 1840 a greater ability to concentrate their attention on a few key areas of research than have their colleagues in less completely quantified fields. In the same period, if I am right, physicists would prove to have been more successful than most other scientists in decreasing the length of contro­versies about scientific theories and in increasing the strength of the consensus that emerged from such controversies. In short, I believe that the nineteenth- century mathematization of physical science produced vastly refined profes­sional criteria for problem selection and that it simultaneously very much increased the effectiveness of professional verification procedures/4 These are, of course, just the changes that the discussion in Section IV would lead us to expect. A critical and comparative analysis of the development of physics during the past century-and-a-quarter should provide an acid test of those conclusions.

Pending tha t test, can we conclude anything at all? I venture the following paradox: The full and intimate quantification of any science is a consumma­tion devoutly to be wished. Nevertheless, it is not a consummation tha t can effectively be sought by measuring. As in individual development, so in the scientific group, m aturity comes most surely to those who know how to wait.

M S. F . Mason, M ain C urrents o f Scientific Icction, note the esoteric quantitative discrcp- T hought (N ew Y ork, 1956), pp. 352-363, pro- ancics which isolated the three problem s— vides an excellent brief sketch of these insti- photoelectric effect, black body radiation, andtutional changes. Much additional m aterial is specific heats—tliat gave rise to quantum me-scattered through, J . T . M crz, H isto ry o f chanics. F or the new effectiveness of verifica-European Thought in the N ineteenth C entury, tion procedures, note the speed w ith which thisvol. I (London, 1923). radical new theory w as adopted by the pro-

c* F o r an exam ple of effective problem se- fession.

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A P P E N D IX

Reflecting on the other papers and on the discussion tha t continued through­out the conference, two additional points tha t had reference to my own paper seem worth recording. Undoubtedly there were others as well, but my mem­ory has proved more than usually unreliable. Professor Price raised the first point, which gave rise to considerable discussion. The second followed from an aside by Professor Spengler, and I shall consider iLs consequences first.

Professor Spengler expressed great interest in my concept of “crises” in the development of a science or of a scientific specialty, but added tha t he had had difficulty discovering more than one such episode in the development of economics. This raised for me the perennial, but perhaps not very im portant question about whether or not the social sciences are really sciences a t all. Though I shall not even attem pt to answer it in that form, a few further re­m arks about the possible absence of crises in the development of a social science may illuminate some part of what is a t issue.

As developed in Section IV, above, the concept of a crisis implies a prior unanimity of the group tha t experiences one. Anomalies, by definition, exist only with respect to firmly established expectations. Experiments can create a crisis by consistently going wrong only for a group that has previously ex­perienced everything’s seeming to go right. Now, as my Sections II and III should indicate quite fully, in the mature physical sciences most things gen­erally do go right. The entire professional community can therefore ordinarily agree about the fundamental concepts, tools, and problems of its science. W ithout that professional consensus, there would be no basis for the sort of puzzle-solving activity in which, as 1 have already urged, most physical scien­tists are normally engaged. In the physical sciences disagreement about fun­damentals is, like the search for basic innovations, reserved lor periods of crisis.06 I t is, however, by no means equally clear that a consensus of anything like similar strength and scope ordinarily characterizes the social scicnces. Experience with my university colleagues and a fortunate year spent at the Center for Advanced Study in the Behavioral Sciences suggest that the funda­mental agreement which physicists, say, can normally take for granted has only recently begun to emerge in a few areas of social-science research. Most other areas are still characterized by fundamental disagreements about the definition of the field, its paradigm achievements, and its problems. While tha t situation obtains (as it did also in earlier periods of the development of the various physical sciences), either there can be no crises or there can never be anything else.

Professor Price’s point was very different and far more historical. H e sug­gested, and I think quite rightly, that my historical epilogue failed to call attention to a very important change in the attitude of physical scientists to­wards measurement tha t occurred during the Scientific Revolution. In com-

451 have developed some o ther significant T h e T h ird (1959) U niversity o f Utah R e- concom itants of th is professional consensus in search Conference on the Identification o f my paper, "T he Essential Tension: T rad ition Creative Scientific Talent (U niversity of U tah and Innovation in Scientific R esearch." T h a t P ress, 1959), pp. 162-177. paper appears in, Calvin W . T ay lo r (cd .),

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meriting on Dr. Crombie’s paper, Price had pointed out tha t not until the late sixteenth century did astronomers begin to record continuous series of obser­vations of planetary position. (Previously they had restricted themselves to occasional quantitative observations of special phenomena.) Only in that same late period, he continued, did astronomers begin to be critical of their quan­titative data, recognizing, for example, that a recorded celestial position is a clue to an astronomical fact rather than the fact itself. When discussing my paper, Professor Price pointed to still other signs of a change in the attitude towards measurement during the Scientific Revolution. For one thing, he emphasized, many more numbers were recorded. More important, perhaps, people like Boyle, when announcing laws derived from measurement, began for the first time to record their quantitative data, whether or not they per­fectly fit the law, rather than simply stating the law itself.

I am somewhat doubtful that this transition in attitude towards numbers proceeded quite so far in the seventeenth century as Professor Price seemed occasionally to imply. Hooke, for one example, did not re}>ort the numbers from which he derived his law of elasticity; no concept of “significant figures” seems to have emerged in the experimental physical sciences before the nine­teenth century. But I cannot doubt that the change was in process and that it is very important. At least in another sort of paper, it deserves detailed examination which I very much hope it will get. Pending that examination, however, let me simply point out how very closely the development of the phenomena emphasized by Professor Price fits the pattern I have already sketched in describing the effects of seventeenth-century Baconianism.

In the first place, except perhaps in astronomy, the seventeenth-century change in attitude towards measurement looks very much like a response to the novelties of the methodological program of the “ new philosophy.” Those novelties were not, as has so often been supposed, consequences of the belief that observation and experiment were basic to science. As Crombie has bril­liantly shown, that belief and an accompanying methodological philosophy were highly developed during the Middle Ages.*’ Instead, the novelties of method in the “ new philosophy” included a belief that lots and lots of experi­ments would be necessary (the plea for natural histories) and an insistence that all experiments and observations be reported in full and naturalistic de­tail, preferably accompanied by the names and credentials of witnesses. Both the increased frequency with which numbers were recorded and the decreased tendency to round them off are precisely congruent with those more general Baconian changes in the attitude towards experimentation at large.

Furtherm ore, whether or not its source lies in Baconianism, the effective­ness of the seventeenth-century’s new attitude towards numbers developed in very much the same way as the effectiveness of the other Baconian novelties discussed in my concluding section. In dynamics, as Professor Koyr6 has re­peatedly shown, the new attitude had almost no effect before the later eight­eenth century. The other two traditional sciences, astronomy and optics, were affected sooner by the change, but only in their most nearly traditional parts.

See particularly h is R obert Grosseteste 1700 (O xfo rd , 1953). and the O rigins o f E xperim enta l Science, 1100-

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And the Baconian sciences, heat, electricity, chemistry, etc., scarcely begin to profit from the new attitude until after 1750. Again it is in the work of Black, Lavoisier, Coulomb, and their contemporaries tha t the first truly significant effects of the change arc seen. And the full transformation of physical scicncc due to that change is scarcely visible before the work of Ampere, Fourier, Ohm, and Kelvin. Professor Price has, I think, isolated another very signifi­cant seventeenth-century novelty. But like so many of the other novel a tti­tudes displayed by the “new philosophy,” the significant effects of this new attitude towards measurement were scarcely manifested in the seventeenth century at all.